A function $g$ satisfies the property that $g(k) = 3k + 5$ for each of the 15 integer values of $k$ in $[1,15]$.
For each statement below, state if it is true or false.
(i) If $g(x)$ is a linear polynomial, then $g(x) = 3x + 5$.
(ii) $g$ cannot be a polynomial of degree 10.
(iii) $g$ cannot be a polynomial of degree 20.
(iv) If $g$ is differentiable, then $g$ must be a polynomial.

10. Let $f ( x )$ and $g ( x ) = x ^ { 3 } + x ^ { 2 } - 2$ be real coefficient polynomials with a common factor of degree greater than 0. Which of the following statements are correct?
(1) $g ( x ) = 0$ has exactly one real root
(2) $f ( x ) = 0$ must have a real root
(3) If $f ( x ) = 0$ and $g ( x ) = 0$ have a common real root, then this root must be 1
(4) If $f ( x ) = 0$ and $g ( x ) = 0$ have a common real root, then the greatest common divisor of $f ( x )$ and $g ( x )$ is a linear polynomial
(5) If $f ( x ) = 0$ and $g ( x ) = 0$ have no common real roots, then the greatest common divisor of $f ( x )$ and $g ( x )$ is a quadratic polynomial

3. Given that $f(x)$ and $g(x)$ are two real-coefficient polynomials, and the remainder when $f(x)$ is divided by $g(x)$ is $x^{4} - 1$. Which of the following options cannot be a common factor of $f(x)$ and $g(x)$?
(1) 5
(2) $x - 1$
(3) $x^{2} - 1$
(4) $x^{3} - 1$
(5) $x^{4} - 1$

Assume $f ( x )$ is a fifth-degree polynomial with real coefficients, and the remainder when $f ( x )$ is divided by $x ^ { n } - 1$ is $r _ { n } ( x )$ , where $n$ is a positive integer. Select the correct options.
(1) $r _ { 1 } ( x ) = f ( 1 )$
(2) $r _ { 2 } ( x )$ is a first-degree polynomial with real coefficients
(3) The remainder when $r _ { 4 } ( x )$ is divided by $x ^ { 2 } - 1$ equals $r _ { 2 } ( x )$
(4) $r _ { 5 } ( x ) = r _ { 6 } ( x )$
(5) If $f ( - x ) = - f ( x )$ , then $r _ { 3 } ( - x ) = - r _ { 3 } ( x )$

In this question, $f ( x )$ is a non-constant polynomial, and $g ( x ) = x f ^ { \prime } ( x )$ $f ( x ) = 0$ for exactly $M$ real values of $x$. $g ( x ) = 0$ for exactly $N$ real values of $x$. Which of the following statements is/are true? I It is possible that $M < N$ II It is possible that $M = N$ III It is possible that $M > N$
A none of them B I only C II only D III only E I and II only F I and III only G II and III only H I, II and III

$P(x)$ and $Q(x)$ are non-constant polynomials, and $R(x)$ is a first-degree polynomial, where
$$P(x) = Q(x) \cdot R(x)$$
the equality is satisfied.
Accordingly, I. The constant terms of polynomials $P(x)$ and $R(x)$ are the same. II. If the graph of $P(x)$ is a parabola, then the graph of $Q(x)$ is a line. III. Every root of polynomial $Q(x)$ is also a root of polynomial $R(x)$. Which of the following statements are always true?
A) Only II
B) Only III
C) I and II
D) I and III
E) II and III